Question

The position of a train as it travels between two stations is described by the equation f(t) = 90t^2 + 2t, where f(t) is the distance in miles and t is time in minutes. What is its average velocity between minutes 3 and 7?

Answer 1

To find the average velocity of the train between minutes \( t = 3 \) and \( t = 7 \), we can use the formula for average velocity over an interval, which is:

\[
\text{Average velocity} = \frac{f(b) - f(a)}{b - a}
\]

where \( f(t) \) is the position function, \( a = 3 \), and \( b = 7 \).

First, we need to calculate \( f(3) \) and \( f(7) \):

1. Calculate \( f(3) \):
\[
f(3) = 90(3^2) + 2(3)
\]
\[
= 90(9) + 6
\]
\[
= 810 + 6
\]
\[
= 816 \text{ miles}
\]

2. Calculate \( f(7) \):
\[
f(7) = 90(7^2) + 2(7)
\]
\[
= 90(49) + 14
\]
\[
= 4410 + 14
\]
\[
= 4424 \text{ miles}
\]

Now, we can find the average velocity:

\[
\text{Average velocity} = \frac{f(7) - f(3)}{7 - 3}
\]
\[
= \frac{4424 - 816}{4}
\]
\[
= \frac{3608}{4}
\]
\[
= 902 \text{ miles per minute}
\]

Therefore, the average velocity of the train between minutes 3 and 7 is \( 902 \) miles per minute.